Periodic-end Dirac operators and Seiberg-Witten theory Tomasz Mrowka - PowerPoint PPT Presentation

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Periodic-end Dirac operators and Seiberg-Witten theory ∗ Tomasz Mrowka 1 Daniel Ruberman 2 Nikolai Saveliev 3 1 Department of Mathematics Massachusetts Institute of Technology 2 Department of Mathematics Brandeis University 3 Department of Mathematics University of Miami Conference on Spectral Geometry, Potsdam, May 2008 ∗ Preliminary report

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC The simplest smooth 4-manifolds Simply connected: S 4 , CP 2 , S 2 × S 2 . Non-simply connected: S 1 × S 3 . Will concentrate on invariants of manifolds X with the homology of S 1 × S 3 . Classical Z 2 -valued invariant ρ ( X ) arising from Rohlin’s signature theorem. Choose oriented M 3 ⊂ X generating H 3 ( X ) . Choose spin 4-manifold W with ∂ W = M ρ ( X ) = ρ ( M ) = 1 8 σ ( W )

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Long-term goal to find Z -valued lift of ρ ( X ) . Applications to classification of manifolds. Applications to homology cobordism and triangulation of high-dimensional manifolds. Approach is to calculate ρ ( X ) analytically via gauge theory–Yang Mills and Seiberg-Witten theory.

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Seiberg-Witten equations Seiberg-Witten theory assigns to a 4-manifold Y and Spin c structure s , a number SW ( Y , s ) , by counting irreducible solutions (up to gauge equivalence) to the Seiberg-Witten equations. Variables: Spin c connection A , spinor ψ ∈ C ∞ ( S + ) , and r ∈ R + � | ψ | 2 = 1 D + A ( g ) ψ = 0 Y F + A + r 2 q ( ψ ) = µ where g is a metric on Y , and µ ∈ Ω 2 + ( Y ; i R ) . A solution is irreducible if r � = 0.

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Seiberg-Witten equations Equations depend on metric on Y and 2-form µ . Generic perturbation µ makes moduli space smooth, oriented 0-manifold. Version of equations with r yield ‘blown-up’ moduli space of Kronheimer-Mrowka. Count irreducible ( r � = 0) solutions to µ -perturbed Seiberg-Witten equations. Independent from g and µ if b + 2 Y > 1. Specialize to case of X with homology of S 1 × S 3 , and write µ = d + β . The algebraic count of irreducible solutions is denoted SW ( X , g , β ) .

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Seiberg-Witten equations Key problem: SW ( X , g , β ) depends on g and β . Consider SW ( X , g t , β t ) for 1-parameter family ( g t , β t ) . Since b + 2 ( X ) = 0, may have solutions ( A t , r t , ψ t ) with r t → 0 as t → t 0 , so count can change. Want some other metric-dependent term with similar jump. For X = S 1 × M 3 , done by Chen (1997) and Lim (2000). Counter-term from η -invariants of Dirac operator and signature operator on M 3 .

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Periodic Dirac operators Proposed counter-term in non-product case: Index of Periodic-end Dirac operator. Setup: Closed spin manifold X with a map f : X → S 1 , surjective on π 1 . This gives X → X , and lift t : ˜ X → R of f . Connected Z -cover ˜ D + : C ∞ (˜ Dirac operator ˜ S + ) → C ∞ (˜ S − ) . For any regular value θ ∈ S 1 for f , a submanifold f − 1 θ = M ⊂ X . D + a Fredholm operator? Question: When is ˜

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Periodic Dirac operators To make sense of this, need to complete C ∞ S ± ) in some 0 (˜ norm. Pick δ ∈ R , and define � e t δ | s | 2 < ∞} L 2 S ± ) = { s | δ (˜ X ˜ as well as Sobolev spaces L 2 k , δ (˜ S ± ) . Should really ask if the dimensions of the kernel/cokernel of D + : L 2 ˜ k , δ (˜ S ± ) → L 2 k − 1 , δ (˜ S ± ) D + is Fredholm on L 2 are finite. If so we’ll be sloppy and say ˜ δ . The most useful case for us is δ = 0.

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Periodic Dirac operators Taubes’ idea: Fourier-Laplace transform ∞ s µ e µ t ( x ) e µ n s ( x + n ) for µ ∈ C s ⇒ ˆ � n = −∞ converts to family of problems on compact X . For each c ∈ C , have the twisted Dirac operator D + c : C ∞ ( S + ) → C ∞ ( S − ) given by D + c s = D + s − log ( c ) dt · s . Theorem 1 (Taubes, 1987) c = { 0 } for all c ∈ C ∗ with Fix δ ∈ R . Suppose that ker D + D + is Fredholm on L 2 2 . Then ˜ | c | = e δ δ .

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Periodic Dirac operators Theorem 2 (R-Saveliev, 2006) D + is Fredholm on L 2 . For a generic metric on X, the operator ˜ Suffices to find one metric with D c invertible ∀ c ∈ S 1 . We apply technique of Ammann-Dahl-Humbert (2006). Invertibility of D c , ∀ c ∈ S 1 , can be pushed across a cobordism.

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC End-periodic manifolds End-periodic manifolds are periodic in finitely many directions, each modeled on a Z covering ˜ X → X . Let M ⊂ X be non-separating; it lifts to a compact submanifold M 0 ⊂ ˜ X . M 0 X ˜ X 0 ˜ M X

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC End-periodic manifolds Let ˜ X 0 be everything to the right of M 0 , and choose a compact oriented spin manifold W with (oriented) boundary − M . From these pieces, form the end-periodic manifold with end modeled X : on ˜ Z = W ∪ M 0 ˜ X 0 M 0 W X 0 ˜

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC End-periodic manifolds Excision principle: Everything we said about Dirac operators on ˜ X holds for Dirac operators on Z . For metric g on X , extending to metric on Z , get Dirac operator D + ( Z , g ) and twisted version D + β ( Z , g ) for β ∈ Ω 1 ( X ; i R ) . Fredholm on L 2 for generic ( g , β ) . ind ( D + β ( Z , g )) depends on choice of W in simple way. Unlike compact case, ind ( D + β ( Z , g )) depends on ( g , β ) . Can jump in family g t if ker ( D + c ( X , g 0 )) � = { 0 } for c ∈ S 1 .

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Observation: ind ( D + β ( Z , g )) jumps at the same place as SW ( X , g , β ) . This suggests that we try to use one to balance the other. Have to get rid of dependence of ind ( D + β ( Z , g )) on compact manifold W . Provisional definition: Consider the quantity β ( Z , g )) − 1 λ SW ( X , g , β ) = SW ( X , g , β ) − ind ( D + 8sign ( W ) Remark: Previous work (R-Saveliev 2004) defines λ Don ( X ) by counting flat connections. Conjecture 3 λ SW ( X , g ) is metric-independent and equals λ Don ( X ) .

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Will discuss approach to independence part of Conjecture 3 shortly. Properties of λ SW Independence from various choices 1 Choice of slice M ⊂ X and lift M 0 ⊂ ˜ X . Choice of W with ∂ W = M , and extension of metric over W . Reduction mod 2 of λ SW is classical Rohlin invariant ρ ( X ) . 2 Item 1: excision principle. Item 2: two ingredients. Involution in Seiberg-Witten theory makes SW ( X , g ) even, and quaternionic nature of Dirac operator makes ind ( D + ( Z , g )) even.

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC Have seen that in a family g t , the invariants SW ( X , g t , β t ) and ind ( D + β t ( Z , g t )) jump at the same t . Change in SW ( X , g , β ) understood: wall-crossing phenomenon in gauge theory. If X = S 1 × M 3 , then change in index is ‘spectral flow’ of Dirac operators on M , studied by Atiyah-Patodi-Singer. Conjecture 3 proved in this situation independently by Chen and Lim. General periodic case more subtle; there’s no operator on M or spectrum to flow.

Seiberg-Witten invariant of X Introduction Seiberg-Witten theory Periodic manifolds and Dirac operators PSC What we know so far: Somewhat easier, but basically equivalent to fix metric g , and vary the exponential weight. Consider fixed operators D + on L 2 δ as δ runs over the interval [ δ 0 , δ 1 ] . When Fredholm, denote its index by ind δ ( D + ) . Denote by S ( δ 0 , δ 1 ) the set of z ∈ C with ker ( D z ) � = 0 and e δ 0 / 2 < | z | < e δ 1 / 2 . By Taubes’ theorem 1, this is a finite set. To each z ∈ S ( δ 0 , δ 1 ) , we associate a ‘multiplicity’ d ( z ) . Definition of d ( z ) complicated; count of solutions to some system of equations. But we can show Lemma 4 If dim ker ( D + z ) = 1 , then d ( z ) = 1 .